Complexity Lower Bounds Using Linear Algebra - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Complexity Lower Bounds Using Linear Algebra write by Satyanarayana V. Lokam. This book was released on 2009-07-20. Complexity Lower Bounds Using Linear Algebra available in PDF, EPUB and Kindle. We survey several techniques for proving lower bounds in Boolean, algebraic, and communication complexity based on certain linear algebraic approaches. The common theme among these approaches is to study robustness measures of matrix rank that capture the complexity in a given model. Suitably strong lower bounds on such robustness functions of explicit matrices lead to important consequences in the corresponding circuit or communication models. Many of the linear algebraic problems arising from these approaches are independently interesting mathematical challenges.
Lower Bounds in Communication Complexity
Lower Bounds in Communication Complexity - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Lower Bounds in Communication Complexity write by Troy Lee. This book was released on 2009. Lower Bounds in Communication Complexity available in PDF, EPUB and Kindle. The communication complexity of a function f(x, y) measures the number of bits that two players, one who knows x and the other who knows y, must exchange to determine the value f(x, y). Communication complexity is a fundamental measure of complexity of functions. Lower bounds on this measure lead to lower bounds on many other measures of computational complexity. This monograph surveys lower bounds in the field of communication complexity. Our focus is on lower bounds that work by first representing the communication complexity measure in Euclidean space. That is to say, the first step in these lower bound techniques is to find a geometric complexity measure, such as rank or trace norm, that serves as a lower bound to the underlying communication complexity measure. Lower bounds on this geometric complexity measure are then found using algebraic and geometric tools.
Geometry and Complexity Theory
Geometry and Complexity Theory - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Geometry and Complexity Theory write by J. M. Landsberg. This book was released on 2017-09-28. Geometry and Complexity Theory available in PDF, EPUB and Kindle. Two central problems in computer science are P vs NP and the complexity of matrix multiplication. The first is also a leading candidate for the greatest unsolved problem in mathematics. The second is of enormous practical and theoretical importance. Algebraic geometry and representation theory provide fertile ground for advancing work on these problems and others in complexity. This introduction to algebraic complexity theory for graduate students and researchers in computer science and mathematics features concrete examples that demonstrate the application of geometric techniques to real world problems. Written by a noted expert in the field, it offers numerous open questions to motivate future research. Complexity theory has rejuvenated classical geometric questions and brought different areas of mathematics together in new ways. This book will show the beautiful, interesting, and important questions that have arisen as a result.
Linear Algebraic Techniques in Algorithms and Complexity
Linear Algebraic Techniques in Algorithms and Complexity - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Linear Algebraic Techniques in Algorithms and Complexity write by Joshua H. Alman. This book was released on 2019. Linear Algebraic Techniques in Algorithms and Complexity available in PDF, EPUB and Kindle. We develop linear algebraic techniques in algorithms and complexity, and apply them to a variety of different problems. We focus in particular on matrix multiplication algorithms, which have surprisingly fast running times and can hence be used to design fast algorithms in many settings, and matrix rank methods, which can be used to design algorithms or prove lower bounds by analyzing the ranks of matrices corresponding to computational tasks. First, we study the design of matrix multiplication algorithms. We define a new general method, called the Universal Method, which subsumes all the known approaches to designing these algorithms. We then design a suite of techniques for proving lower bounds on the running times which can be achieved by algorithms using many tensors and the Universal Method. Our main limitation result is that a large class of tensors generalizing the Coppersmith-Winograd tensors (the family of tensors used in all record-holding algorithms for the past 30+ years) cannot achieve a better running time for multiplying n by n matrices than O(n2[superscript .]168). Second, we design faster algorithms for batch nearest neighbor search, the problem where one is given sets of data points and query points, and one wants to find the most similar data point to each query point, according to some distance measure. We give the first subquadratic time algorithm for the exact problem in high dimensions, and the fastest known algorithm for the approximate problem, for various distance measures including Hamming and Euclidean distance. Our algorithms make use of new probabilistic polynomial constructions to reduce the problem to the multiplication of low-rank matrices. Third, we study rigid matrices, which cannot be written as the sum of a low rank matrix and a sparse matrix. Finding explicit rigid matrices is an important open problem in complexity theory with applications in many different areas. We show that the Walsh-Hadamard transform, previously a leading candidate rigid matrix, is in fact not rigid. We also give the first nontrivial construction of rigid matrices in a certain parameter regime with applications to communication complexity, using an efficient algorithm with access to an NP oracle.
Geometry and Complexity Theory
Geometry and Complexity Theory - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Geometry and Complexity Theory write by J. M. Landsberg. This book was released on 2017-09-28. Geometry and Complexity Theory available in PDF, EPUB and Kindle. This comprehensive introduction to algebraic complexity theory presents new techniques for analyzing P vs NP and matrix multiplication.
